Topic
JSJ decomposition
About: JSJ decomposition is a research topic. Over the lifetime, 89 publications have been published within this topic receiving 2320 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, the authors provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics, for n ≥ 3.
Abstract: We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n≥3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is algorithmically constructible.
88 citations
TL;DR: In this article, it was shown that any two JSJ decompositions of a group are related by an elementary deformation, and that strongly slide-free (or non-sliding)-JSJ decomposition is unique.
Abstract: We give an example of two JSJ decompositions of a group that are not related by conjugation, conjugation of edge-inclusions, and slide moves. This answers the question of Rips and Sela stated in (RS). On the other hand we observe that any two JSJ decompositions of a group are related by an elementary deformation, and that strongly slide-free JSJ decompositions are genuinely unique. These results hold for the decompositions of Rips and Sela, Dunwoody and Sageev, and Fujiwara and Papasoglu, and also for accessible decompositions. AMS Classification 20F65; 20E08, 57M07
66 citations
Posted Content•
TL;DR: In this article, the authors give a general simple definition of JSJ decompositions by means of a universal maximality property, and prove that JSJ-decomposition of finitely presented groups always exist.
Abstract: This paper and its companion arXiv:1002.4564 have been replaced by arXiv:1602.05139.
We give a general simple definition of JSJ decompositions by means of a universal maximality property. The JSJ decomposition should not be viewed as a tree (which is not uniquely defined) but as a canonical deformation space of trees. We prove that JSJ decompositions of finitely presented groups always exist, without any assumption on edge groups. Many examples are given.
53 citations
TL;DR: The relation between the stable complexity of a closed manifold and Gromov's simplicial volume has been studied in this article, where it is shown that the latter is smaller than the former.
Abstract: Let the complexity of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree we can promote it to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which call the "stable complexity" of M.
We study here the relation between the stable complexity of M and Gromov's simplicial volume ||M||. It is immediate to show that ||M|| is smaller or equal than the stable complexity of M and it is natural to ask whether the two quantities coincide on aspherical manifolds with residually finite fundamental group. We show that this is not always the case: there is a constant C_n<1 such that ||M|| is smaller than C_n times the stable complexity for any hyperbolic manifold M of dimension at least 4.
The question in dimension 3 is still open in general. We prove that the stable complexity equals ||M|| for any aspherical irreducible 3-manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figure-eight knot complement. The equality holds for all closed hyperbolic 3-manifolds if a particular three-dimensional version of the Ehrenpreis conjecture is true.
46 citations
TL;DR: In this paper, the authors define a tree of cylinders T_c which is invariant under automorphisms of a finitely generated group if T is a JSJ splitting, and obtain Out(G)-invariant cyclic or abelian JSJ splittings.
Abstract: Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T_c. This tree only depends on the deformation space of T; in particular, it is invariant under automorphisms of G if T is a JSJ splitting. We thus obtain Out(G)-invariant cyclic or abelian JSJ splittings. Furthermore, T_c has very strong compatibility properties (two trees are compatible if they have a common refinement).
44 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 2 |
| 2020 | 5 |
| 2019 | 5 |
| 2018 | 5 |
| 2017 | 5 |
| 2016 | 10 |